MTH102: Elementary Mathematics Ii TMA4 ANSWER TO MTH102 TMA 4

MTH102: Elementary Mathematics Ii TMA4

Instructions: Answer All .
After Submission, you will not be able to answer these questions

Question 1 : Find the volume of a sphere generated by a semicircle $$y=\sqrt{(}{r}^2-x^2)$$

revolving around the x-axis
$$4\pi \frac{r^3}{3}$$
$$\pi \frac{r^3}{4}$$
$$4\pi \frac{r^3}{2}$$
$$-\pi \frac{-r^3}{2}$$

---

Question 2 : Evaluate $$\int x^2{e}^{3x}dx$$
$$\frac{e^x}{3}\left(x^2+\frac{2x}{3}-\frac{2}{5}\right)+c$$
$$-\frac{e^{3x}}{3}\left(x^2+\frac{2x}{3}-\frac{2}{9}\right)+c$$
$$\frac{e^{3x}}{3}\left(x^2-\frac{2x}{3}+\frac{2}{9}\right)+c$$
$$\frac{e^{2x}}{3}\left(x^3-\frac{x}{4}+\frac{2}{9}\right)+c$$

---

Question 3 : Evaluate $$\int \frac{x+1}{x^2-3x+2}dx$$
$$3ln(x-2)+2ln(x+1)+c$$
$$3ln(x+2)-2ln(x+1)+c$$
$$-3ln(x-2)-2ln(x-1)+c$$
$$3ln(x-2)-2ln(x-1)+c$$

---

Question 4 : Evaluate the limit $$\underset{h\to 0}{lim}\frac{2(-3+h{)}^2-18}{h}$$
6
8
14
12

---

Question 5 : Evaluate the limit $$limx\to \infty \frac{2x^4-x^2+8x}{-5x^4+7}$$
$$\frac{1}{2}$$
$$\frac{2}{3}$$
$$\frac{1}{3}$$
$$\frac{3}{4}$$

---

Question 6 : Find $$\int x\cos a{x}^{2}dx$$ with respect to x
$$\sin 2x+c$$
$$\frac{1}{2}a\sin a{x}^2+c$$
$${\sec }^{2}x+1$$
$${\cos }^{3}x+c$$

---

Question 7 : Evaluate the limit $$limx\to -\infty \frac{x^2-5t-9}{2x^4+3x^3}$$
1
0
2
4

---

Question 8 : Evaluate $$\int e^{4x}dx$$
$$\frac{1}{4}{e}^{4x}+c$$
$$3e^{\frac{x}{3}}+c$$
$$ex+c$$
$$\frac{1}{3}ex+c$$

---

Question 9 : Find the $$\int {\tan }^{3}x{\sec }^{3}xdx$$
$${\tan }^{2}x+1$$
$${\cot }^{2}x+1$$
$${\sec }^{2}x+1$$
$${\sec }^{2}x$$

---

Question 10 : Given $$\frac{2x^5+x^2-5}{t^2}$$, find $$\frac{dy}{dx}$$ by using the first principle
c$$-t^{-2}+8t^{-3}$$
$$6t+7t^{-3}$$
$$6t^2+10t^{-3}$$
$$t^2+5t^{-3}$$

---

Question 11 : Given $$y\left(x\right)=x^4-4x^3+3x^2-5x$$, evaluate $$\frac{d^{4}y}{d{x}^4}$$
22
24
30
42

---

Question 12 : Evaluate the limit $$limt\to 4\frac{t-\sqrt{(}3+4)}{4-t}$$
$$\frac{-3}{8}$$
$$\frac{3}{4}$$
$$\frac{-1}{8}$$
$$\frac{-5}{8}$$

---

Question 13 : Evaluate $$\int x{e}^{6x}dx$$
$$\frac{x}{3}{e}^{6x}+1\frac{1}{6}{e}^{6x}+c$$
$$\frac{x}{6}{e}^{6x}+1\frac{1}{36}{e}^{6x}+c$$
$$-\frac{x}{6}{e}^{6x}+1\frac{1}{36}{e}^{6x}+c$$
$$\frac{x}{6}{e}^{6x}-1\frac{1}{36}{e}^{6x}+c$$

---

Question 14 : Given f(x) = $$(7x^4-5x^3)$$, evaluate $$\frac{df\left(x\right)}{dx}$$
$$2x^3-15x^2$$
$$28x^2-15x^2$$
$$28x^3-15x^2$$
$$7x^4-5x^3$$

---

Question 15 : Differentiate with respect to x: $$f\left(x\right)=(ax{}^3+bx)$$
$$a{x}^2+b$$
$$3x^2+1$$
$$3a{x}^2+b$$
$$3a-b$$

---

Question 16 : Differentiate $$y=3\sqrt{(}{x}^2\left)\right(2x-x^2)$$ with respect to x
$$y=\frac{10x^{\frac{2}{3}}}{3}+\frac{8x^{\frac{5}{3}}}{3}$$
$$y=\frac{5x^{\frac{2}{3}}}{3}+\frac{4x^{\frac{5}{3}}}{3}$$
$$y=\frac{10x^{\frac{2}{3}}}{3}-\frac{8x^{\frac{5}{3}}}{3}$$
$$y=\frac{5x^{\frac{2}{3}}}{3}-\frac{4x^{\frac{5}{3}}}{3}$$

---

Question 17 : Determine $$\int \frac{x^2+1}{(x+2{)}^3}$$
$$ln(x+2)-\frac{4}{x+2}-\frac{5}{2(x+3{)}^2}+c$$
$$-ln(x+2)-\frac{4}{x+2}-\frac{5}{2(x+3{)}^2}+c$$
$$ln(x-2)+\frac{4}{x-2}-\frac{5}{2(x+3{)}^2}+c$$
$$ln(x+2)+\frac{4}{x+2}-\frac{5}{2(x+3{)}^2}+c$$

---

Question 18 : Evaluate the limit $$limx\to \infty \frac{6e^{4x}-e^{-2x}}{8e^{4x}-e^{2x}+3e^{-x}}$$
$$\frac{3}{5}$$
$$\frac{1}{4}$$
$$\frac{1}{2}$$
$$\frac{3}{4}$$

---

Question 19 : Find $$\int {\sec }^{3}x\tan xdx$$
$$\sin 2x+c$$
$$\cos 2x+c$$
$$\frac{{\cos }^{3}x}{2}+c$$
$$\frac{{\sec }^{3}x}{3}+c$$

---

Question 20 : Find the derivative $$f\left(x\right)=2x^2-16x+35$$ by using first principle
$$x+16$$
$$3x-5$$
$$2x-8$$
$$4x-16$$

---